Problem 9 - Algebra for College Students

Question

Use the graphing approach to determine whether the system is consistent, the system in inconsistent, or the equations are dependent. If the system is consistent, find the solution set from the graph and check it. \(\left(\begin{array}{r}x-\frac{y}{2}=-4 \\ 8 x-4 y=-1\end{array}\right)\)

Step-by-Step Solution

Answer

The system is inconsistent; the equations are parallel and do not intersect.

Step 1: Convert Equations to Slope-Intercept Form

For the graphing approach, start by converting both equations to the slope-intercept form, which is \( y = mx + b \). For the first equation: \( x - \frac{y}{2} = -4 \), solve for \( y \) to get \( y = 2x + 8 \). For the second equation: \( 8x - 4y = -1 \), solve for \( y \) to get \( y = 2x + \frac{1}{4} \).

Step 2: Analyze the Slopes and Intercepts

Both equations have a slope of 2, indicating that they are parallel. However, they have different y-intercepts (8 for the first and \( \frac{1}{4} \) for the second). Parallel lines with different y-intercepts imply that the lines never intersect and are therefore inconsistent.

Step 3: Conclusion Based on Graph Analysis

Since the lines are parallel and do not intersect, the system is inconsistent. This means there is no solution for the system of equations. They do not have any points in common.

Key Concepts

Graphing Method

The graphing method is a visual way to solve systems of equations. By graphing each equation on the same set of axes, you can determine whether they intersect, are parallel, or coincide. The graphing method provides a clear picture of the relationships between the equations.
Start by converting each equation to the slope-intercept form. This makes it easier to graph the lines and understand the relationship between them. In our system of equations, we begin by rewriting each in the form of \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
Using graphing:

• Graph each line on the same set of axes.
• Look for points where the lines intersect (if any).
• Determine the type of system (consistent, inconsistent, or dependent) based on the graph observations.

Graphing is especially useful for understanding systems of equations visually and getting insights into how the equations relate.

Slope-Intercept Form

The slope-intercept form of a linear equation is \( y = mx + b \). This form is very useful in graphing because it easily shows you the slope and y-intercept of the line.
The slope, \( m \), tells you how steep the line is and the direction it goes. It represents the change in \( y \) for a one-unit increase in \( x \). The y-intercept, \( b \), is the point where the line crosses the y-axis.

• To convert an equation to slope-intercept form, solve for \( y \) in terms of \( x \).
• This form makes graphing straightforward by giving you two key pieces of information: slope \( m \) and y-intercept \( b \).

The first equation \( x - \frac{y}{2} = -4 \) converts to \( y = 2x + 8 \), revealing a slope of 2 and a y-intercept of 8.
The second equation \( 8x - 4y = -1 \) converts to \( y = 2x + \frac{1}{4} \), also showing a slope of 2 but a y-intercept of \( \frac{1}{4} \). The matching slopes indicate the lines are parallel.

Parallel Lines

Parallel lines are lines in the same plane that never intersect. They have the same slope but different y-intercepts. When graphing, if two lines have exactly the same slope, they are parallel.
In our system of equations:

• Both lines have a slope of 2.
• However, the y-intercepts differ: 8 for the first line and \( \frac{1}{4} \) for the second.

Because the slopes are equal and the y-intercepts are different, these lines are parallel. As they never meet, there is no solution to the system of equations. Parallel lines visually demonstrate that the equations cannot simultaneously be true for any point in the coordinate plane.

Inconsistent System

An inconsistent system of equations is one that has no solution. This occurs when the equations represent parallel lines, which means they will never intersect.

• Parallel lines with the same slope but different y-intercepts lead to inconsistency.
• An inconsistent system means there is no common point (or ordered pair) that satisfies both equations.

From our previous steps: the system \( y = 2x + 8 \) and \( y = 2x + \frac{1}{4} \) is inconsistent because the lines do not meet at any point.
When solving systems of equations, identifying parallel lines is key to recognizing inconsistency. No shared solution exists.